Reflecting cardinals

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Reflection is a fundamental motivating concern in set theory. The theory of ZFC can be equivalently axiomatized over the very weak Kripke-Platek set theory by the addition of the reflection theorem scheme, below, since instances of the replacement axiom will follow from an instance of $\Delta_0$-separation after reflection down to a $V_\alpha$ containing the range of the defined function. Several philosophers have advanced philosophical justifications of large cardinals based on ideas arising from reflection.

Reflection theorem

Proved by Montague. add material here.

Reflecting cardinals

For any class $\Gamma$ of formulas, a cardinal $\kappa$ is $\Gamma$-reflecting if and only if $H_\kappa\prec_\Gamma V$, meaning that for any $\varphi\in\Gamma$ and $a\in H_\kappa$ we have $V\models\varphi[a]\iff H_\kappa\models\varphi[a]$. (Warning: some authors have used the term $\Sigma_2$-reflecting to imply that the cardinal also is inaccessible, but here on Cantor's attic we shall refer to those as the inaccessible reflecting cardinals.

  • A simple Löwenheim-Skolem argument shows that every infinite cardinal $\kappa$ is $\Sigma_1$-reflecting.
  • For each natural number $n$, the $\Sigma_n$-reflecting cardinals form a closed unbounded proper class of cardinals, as a consequence of the reflection theorem, below.

A cardinal $\kappa$ is reflecting, written $V_\kappa\prec V$, if it is $\Sigma_n$-reflecting for each $n$. This is not expressible by a single assertion in the language of set theory (since if it were, the least such $\kappa$ would have to have a smaller one inside $V_\kappa$ by elementarity). Nevertheless, $V_\kappa\prec V$ is expressible as a scheme in the language of set theory with a parameter (or constant symbol) for $\kappa$.

Although it may be surprising, the existence of a fully reflecting cardinal is equiconsistent with ZFC. This can be seen by a simple compactness argument, using the fact that the theory ZFC+"$\kappa$ is reflecting" is finitely consistent, if ZFC is consistent, precisely by the observation about $\Sigma_n$-reflecting cardinals above.

$\Sigma_2$ reflecting cardinals

The $\Sigma_2$-reflecting cardinals are a particularly useful and robust class of cardinals, because of the following characterization: $\kappa$ is $\Sigma_2$-reflecting if and only if for any $x\in V_\kappa$ and any formula $\varphi$ of any complexity, whenever there is an ordinal $\alpha$ such that $V_\alpha\models\varphi[x]$, then there is $\alpha\lt\kappa$ with $V_\alpha\models\varphi[x]$. The reason this is equivalent to $\Sigma_2$-reflecting is that assertions of the form $\exists \alpha\ V_\alpha\models\varphi(x)$ have complexity $\Sigma_2(x)$, and conversely all $\Sigma_2(x)$ assertions can be made in that form.

It follows, for example, that if $\kappa$ is $\Sigma_2$-reflecting, then any feature of $\kappa$ or any larger cardinal than $\kappa$ that can be verified in a large $V_\alpha$ will reflect below $\kappa$. So if $\kappa$ is $\Sigma_2$ reflecting and measurable, for example, then there must be unboundedly many measurable cardinals below $\kappa$.

The Feferman theory

This is the theory, expressed in the language of set theory augmented with a new unary class predicate symbol $C$, asserting that $C$ is a closed unbounded class of cardinals, and every $\gamma\in C$ has $V_\gamma\prec V$. In other words, the theory consists of the following scheme of assertions: $$\forall\gamma\in C\ \forall x\in V_\gamma\ \bigl[\varphi(x)\iff\varphi^{V_\gamma}(x)\bigr]$$ as $\varphi$ ranges over all formulas. Thus, the Feferman theory asserts that the universe $V$ is the union of a chain of elementary substructures $$V_{\gamma_0}\prec V_{\gamma_1}\prec\cdots\prec V_{\gamma_\alpha}\prec\cdots \prec V$$ Although this may appear at first to be a rather strong theory, since it seems to imply at the very least that each $V_\gamma$ for $\gamma\in C$ is a model of ZFC, this conclusion would be incorrect. In fact, the theory does not imply that any $V_\gamma$ is a model of ZFC, and does not prove $\text{Con}(\text{ZFC})$; rather, the theory implies for each axiom of ZFC separately that each $V_\gamma$ for $\gamma\in C$ satisfies it. Since the theory is a scheme, there is no way to prove from that theory that any particular $\gamma\in C$ has $V_\gamma$ satisfying more than finitely many axioms of ZFC. In particular, a simple compactness argument shows that the Feferman theory is consistent provided only that ZFC itself is consistent, since any finite subtheory of the Feferman theory is true by the reflection theorem in any model of ZFC. It follows that the Feferman theory is actually conservative over ZFC, and proves with ZFC no new facts about sets that is not already provable in ZFC alone.

The Feferman theory was proposed as a natural theory in which to undertake the category-theoretic uses of Grothendieck universes, but without the large cardinal penalty of a proper class of inaccessible cardinals. Indeed, the Feferman theory offers the advantage that the universes are each elementary substructures of one another, which is a feature not generally true under the universe axiom.

Inaccessible reflecting cardinal

A cardinal $\kappa$ is an inaccessible reflecting cardinal if it is inaccessible and reflecting. The existence of such a cardinal is equiconsistent to the assertion ORD is Mahlo.

Maximality Principle

The existence of an inaccessible reflecting cardinal is equiconsistent with the boldface maximality principle $\text{MP}(\mathbb{R})$, which asserts of any statement $\varphi(r)$ with parameter $r\in\mathbb{R}$ that if $\varphi(r)$ is forceable in such a way that it remains true in all subsequent forcing extensions, then it is already true; in short, $\text{MP}(\mathbb{R})$ asserts that every possibly necessary statement with real parameters is already true. Hamkins showed that if $\kappa$ is an inaccessible reflecting cardinal, then there is a forcing extension with $\text{MP}(\mathbb{R})$, and conversely, whenever $\text{MP}(\mathbb{R})$ holds, then there is an inner model with an inaccessible reflecting cardinal.